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Complex Patterns in Oscillatory Systems

Stripes on a mackerel fish [1]

The phenomenon of pattern formation has been observed in many forms,
from wind driven sand ripples, to stripes on the back of mackerel
fish, to tiger fur coloration, to contraction waves in
the heart, the breakup of which is thought to be associated with
fibrillation.

Stripe formation in altocumulus clouds [2]

Stripe formation in sand dunes [3]

I am interested in understanding the formation of complex spatial
patterns in oscillatory media with resonant forcing. Such periodic or
quasi-periodic patterns have been studied and observed experimentally
in Faraday systems where a thin liquid layer is vibrated vertically to
form surface wave patterns like the superlattice patterns below.

Faraday wave patterns [4]

B-Z reaction, unforced (top) and with resonant forcing (bottom) [5].

An example of an oscillatory system is the Belousov-Zhabotinsky (B-Z)
reaction, where oscillations are in chemical
concentration. Experimental observations of the reaction in a thin
layer have shown that the oscillations form travelling waves which can
organize themselves into spiral waves. With the application of
resonant forcing, the oscillations instead form standing waves which
organize themselves in labyrinthine patterns. On the right is a
snapshot from an experiment showing both regimes.

The universal description of small amplitude oscillations is given by
the complex Ginzburg-Landau equation (CGLE) for the amplitude of
oscillations. I extend this equation to include terms that describe
external forcing near integer multiples of the natural or resonant
frequency of oscillation of the system, where A represents the complex
amplitude of oscillations:

At=(μ+iσ)A+(1+iβ)ΔA -(1+iα)A|A|+γ(cos(χ)+eiνtsin(χ))A*+ηA*2

Such forcing induces the formation of spatial patterns in the system
through the combination of standing waves of different orientations.
I am interested in complex periodic or quasi-periodic patterns, which
are formed by the combination of more than three waves. I have shown
that in oscillatory systems patterns with up to 4-fold and 5-fold
rotational symmetries can be made stable through suitable choice of
the forcing function. In regimes where there is competition between
stable patterns, I have used energy arguments to predict the pattern
that will prevail. I then confirmed my predictions through numerical
simulations in large systems. Below are snapshots from numerical
simulations of the CGLE movies showing some of the different patterns;
click on the images to see a movie of their temporal evolution from a
noisy initial condition [6].